1. Introduction to the Normal distribution
The Normal distribution is famous, and we just used it as an example to learn about continuous distributions. We will not go into the story of the Normal distribution in depth,
2. Normal distribution
but for the purposes of this course, it suffices to say that it describes a continuous variable whose PDF is symmetric and has a single peak.
3. Normal distribution
Indeed, that is what its PDF looks like.
The Normal distribution is parametrized by two parameters.
4. Normal distribution
The mean determines
5. Normal distribution
where the center of the peak is.
6. Normal distribution
The standard deviation
7. Normal distribution
is a measure of how wide the peak is, or how spread out the data are.
Note that
8. Normal distribution
the mean and
9. Normal distribution
standard deviation are the names of the parameters of the Normal distribution. Do not confuse these with the mean and standard deviation that we computed directly from the data when doing exploratory data analysis. I know that nomenclature is confusing, but it is important to keep straight.
10. Comparing data to a Normal PDF
Now, if we add a histogram of the Michelson measurements, we see that the measured speed of light in air looks to be Normally distributed. Comparing the histogram to the PDF suffers from binning bias, so it is better to compare the ECDF of the data to the theoretical CDF of the Normal distribution.
11. Checking Normality of Michelson data
To compute the theoretical CDF, we can use rng-dot-normal to draw samples and then we can compute the CDF. As was the case with sampling out of the binomial distribution, we need to provide parameters, in this case the mean and standard deviation, to parametrize the Normal distribution we are sampling out of.
The mean and standard deviations computed from the data are good estimates, so we'll compute them and pass them into the rng-dot-normal function to take our samples.
We then use the function we already wrote to compute the ECDFs of the data and of the Normally distributed theoretical samples we just drew.
12. Checking Normality of Michelson data
Finally, we plot the theoretical and empirical CDFs on the same plot.
13. Checking Normality of Michelson data
With the absence of binning bias, it is much clearer that the Michelson data are approximately Normally distributed.
14. Let's practice!
Now that you can sample out of a Normal distribution, let's practice using it!